Classical covariance control theory is extended to the case of nonlinear systems using the method of statistical linearization. The design procedure is applied to several nonlinear systems of civil engineering interest including hysteretic oscillators. The idea of covariance control is then generalized to the problem of response moment specification where higher order response moments are prescribed with the hope of having more authority over response extremes. The algorithm is then demonstrated by application to a Duffing oscillator.